TITLE: Examples of completions and algebraic closures
QUESTION [7 upvotes]: It is widely known that the algebraic closure of the $p$-adic completion $\mathbb{Q}_p$ of $\mathbb{Q}$ isn't complete anymore. It's completion is complete and known as $\mathbb{C}_p$. 
I have read in a book about non-archimedean analysis that in this case the process ends, which means that $\mathbb{C}_p$ is also algebraically closed.
My question is: is there an example of a field K, in which the algebraic closure $K^{alg}$ isn't complete, and the completion of $K^{alg}$ isn't algebraically closed ? And how do I construct such an example.

REPLY [7 votes]: No, there is not.
If the valuation is archimedean, by Ostrowski the field is isomorphic to the real or complex numbers, so the algebraic closure will already be complete.
If the valuation is non-archimedean, the completion of the algebraic closure will always be algebraically closed. See for example here: http://math.stanford.edu/~conrad/248APage/handouts/algclosurecomp.pdf

REPLY [3 votes]: There is a theorem of Kurschák which asserts that the completion of a valued algebraically closed fied is algebraically closed. This is proved in Paulo Ribenboim's The theory of classical valuations.